Cohomological induction in various categories and the maximal globalization conjecture
This article completes a circle of ideas concerning the representations of semisimple Lie groups. The Dolbeault cohomology of a complex homogeneous space of a Lie group \(G\) with values in a homogeneous vector bundle carries a representation of~\(G\). When \(G\) is compact, the Bott-Borel-Weil theorem identifies this representation. This construction concretely realizes of all irreducible representations of a compact group. When \(G\) is semisimple but non-compact Schmid showed that the discrete series representations are similarly realized on Dolbeault cohomology. For other representations of semisimple groups, however, there are severe technical difficulties in utilizing Dolbeault cohomology. To circumvent these difficulties, Zuckerman introduced cohomological induction, an algebraic analogue of Dolbeault cohomology. Vogan, who studied extensively the Zuckerman construction, introduced the term `maximal globalization conjecture' for the statement that this construction agrees with the corresponding Dolbeault cohomology, at least on the level of Harish-Chandra modules. This conjecture is finally proved in this article. In the case of the discrete series, the inducing representation is finite-dimensional. In an earlier article [J. Funct. Anal. 129, 428-454 (1995; Zbl 0855.22014)] the author proved the maximal globalization conjecture in general when the inducing representation is finite-dimensional. As did Schmid for the discrete series, Wong uses a mapping often referred to nowadays as the Penrose transform. This article treats the general case and it does so by an indirect route. There are five constructions on this circular tour and Wong finally connects the circle by establishing a link between \(G\)-equivariant constructible sheaves and Dolbeault cohomology. Other links in the circle were conjectured by Kashiwara and proved by Kashiwara-Schmid and Mirkovic-Uzawa-Vilonen. In fact, when the inducing representation is infinite-dimensional, there are already some questions as to what one means by a homogeneous vector bundle and its Dolbeault cohomology. Wong answers these questions by writing down an explicit `Dolbeault complex'. He uses functional analysis to establish the technical point that the operator in this complex has closed range. This is the key to an abstract proof of the maximal globalization conjecture. However, he also establishes that an explicit Taylor series map induces the isomorphism claimed in the conjecture. This is a very readable article.
scientific article; zbMATH DE number 1425173
scientific article